# derivative application

• Apr 25th 2009, 10:32 PM
slaypullingcat
derivative application
Find the equation of any quadratics that passes through the origin and are tangent to both y=-2x-4 and y=8x-49.

I have thought about this problem for most of the afternoon now and got no where. I tried drawing it to see the problem but no good. I just don't know where to start or how to approach this problem. What/how should I be thinking?

Thank you
• Apr 26th 2009, 12:57 AM
running-gag
Hi

The quadratics you are looking for passing through the origin, their generic equation is \$\displaystyle f(x) = ax^2 + bx\$

The tangent at a point whose abscissa is \$\displaystyle x_0\$ is :
\$\displaystyle T_0 : y = (2ax_0+b)x-ax_0^2\$

The quadratics are tangent to both y=-2x-4 and y=8x-49 iff there exists \$\displaystyle x_0\$ and \$\displaystyle x_1\$ such that
\$\displaystyle 2ax_0+b = -2\$
\$\displaystyle -ax_0^2 = -4\$
\$\displaystyle 2ax_1+b = 8\$
\$\displaystyle -ax_1^2 = -49\$

Solve for a, b, x0 and x1.